process-induced stress and deformation in thick-section thermoset composite laminates

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1992 26: 626Journal of Composite MaterialsTravis A. Bogetti and John W. Gillespie, Jr

Composite LaminatesProcess-Induced Stress and Deformation in Thick-Section Thermoset

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626

Process-Induced Stress andDeformation in Thick-Section Thermoset

Composite Laminates

TRAVIS A. BOGETTI*

Department of the ArmyBallistic Research LaboratoryAberdeen Proving Ground

MD 21005-5066

JOHN W. GILLESPIE, JR.**Center for Composite Materials

Department of Mechanical EngineeringUniversity of DelawareNewark, DE 19716

(Received June 11, 1990)(Remsed Apnl 2, 1991)

ABSTRACT: A study of process-induced stress and deformation in thick-section

thermosetting composite laminates is presented. A methodology is proposed for predict-ing the evolution of residual stress development during the curing process. A one-dimensional cure simulation analysis is coupled to an incremental laminated plate theorymodel to study the relationships between complex gradients in temperature and degree ofcure, and process-induced residual stress and deformation. Material models are proposedto describe the mechanical properties, thermal and chemical strains of the thermoset resinduring cure. These material models are incorporated into a micromechanics model to pre-dict the effective mechanical properties and process-induced strains of the composite dur-ing cure. Thermal expansion and cure shrinkage contribute to changes in material specificvolume and represent important sources of internal loading included in the analysis. Tem-perature and degree of cure gradients that develop during the curing process represent fun-damental mechanisms that contribute to stress development not considered in traditionalresidual stress analyses of laminated composites. Model predictions of cure dependentepoxy modulus and curvature in unsymmetric graphite/epoxy laminates are correlatedwith experimental data. The effects of processing history (autoclave temperature cure cy-cle), laminate thickness, resin cure shrinkage and laminate stacking sequence on the evo-lution of process-induced stress and deformation in thick-section glass/polyester and

*Mechanical Engineer**Associate Director, Center for Composite Materials, and Associate Professor of Mechanical Engmeenng,

University of Delaware

Journal of COMPOSITE MATERIALS, Vol. 26, No. 5/1992

0021-9983/92/05 0626-35 $6.00/0@ 1992 Technomic Publishmg Co., Inc

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graphite/epoxy laminates during cure are studied. The magnitude of process-induced re-sidual stress is sufficient to mitiate transverse cracks and delammations. The results

clearly indicate that the mechanics and performance of thick-section thermoset laminatesare strongly dependent on processing history.

1. INTRODUCTION

HE DEVELOPMENT OF residual stress is strongly influenced by processinghistory. Residual stresses can have a significant effect on the mechanical per-formance of composite structures by inducing warpage or initiating matrix cracksand delaminations [1-4]. Processing concerns associated with thermosettingcomposites become increasingly important for components of appreciable thick-ness. The most common problem is an increase in temperature resulting from theresin exothermic chemical reaction (polymerization) that may raise internal tem-peratures to levels inducing material degradation. A second concern is the com-plex gradients in temperature and degree of cure accentuated by increased thick-ness. This study focuses on the evolution of macroscopic inplane residual stressesin thick thermoset laminates resulting from complex temperature and degree ofcure gradients that develop during the curing process.

Traditional analyses of residual stresses in thermosetting composite laminatesare based on thermal expansion mismatch between adjacent plies, a uniform tem-perature difference between the cure temperature and ambient and no stress de-velopment prior to completion of the curing process [3-6]. These analyses havebeen quite successful in predicting residual stresses in thin laminates, where auniform through-the-thickness temperature distribution assumption is justified.Such an approach, however, is not appropriate for predicting the process-inducedstresses in thick thermoset composite laminates where complex temperature anddegree of cure gradients develop during the curing process [7-13]. As we demon-strate, these temperature and degree of cure gradients represent important andunique mechanisms that contribute to the development of stress and deformationin thick-section composite laminates.These inherent mechanisms are quite similar to those which govern the devel-

opment of residual stresses encountered in the manufacture of tempered glass[14-17]. Stresses result from the interactions between spatially varying thermalcontractions and viscoelastic material response induced by severe temperaturegradients which develop during the quenching process. Early treatment of thistype of stress development due to viscoelastic material response in the presenceof thermal gradients was based on the time-temperature superposition principleof rheologically simple material behavior [18]. This approach has been applied tostudy the macroscopic inplane residual stress development under the rapid cool-ing process of an isotropic epoxy plate [19] and more recently to the quenchingof thermoplastic matrix composite laminates [20].Complex temperature and degree of cure gradients which develop in thick-

section thermosetting laminates during the curing process induce spatially vary-ing material response similar to that which induces viscoelastic stress develop-ment in rapid cooling and quenching processes. Birefringence patterns haveshown the effect of temperature and degree of cure gradients on the development



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of residual stresses in thick thermoset castings [7]. In a series of papers, Levitskyand Shaffer [21-23] have studied the development of residual stresses includingtemperature gradients and spatially varying chemical hardening effects on me-chanical properties in isotropic thermosetting materials accentuated by the exo-thermic chemical reaction. These studies indicate that temperature and degree ofcure gradients significantly influence stress development prior to a fully curedstate and consequently should be accounted for in studying the evolution ofprocess-induced stress and deformation in thick-section thermosetting compositelaminates.Another important mechanism contributing to stress development not con-

sidered in the analyses cited [21-23] is the volumetric shrinkage of the thermoset-ting resin associated with the cross-link polymerization reaction. Nelson andCairns have demonstrated the importance of volumetric (chemical) shrinkage ondimensional changes in thin angle bend laminates in the absence of cure gradients[24]. The effects of chemical shrinkage on residual stress development in thicklaminates could be potentially worse, for example, if shrinkage were to occurwithin the interior region of a laminate that is constrained by a fully cured ex-terior region. A similar situation occurs on the micromechanics level where resinshrinkage during cure is constrained by the reinforcement phase [25].As a thermosetting resin cures its material characteristics change dramatically,

transitioning from the behavior of a viscous liquid (low shear stiffness), in its un-cured state, to a viscoelastic or elastic solid (high shear stiffness), in its fullycured state. Significant reduction in specific volume (cure shrinkage) associatedwith the cross-link polymerization reaction also occurs during cure. Thesechanges in the resin directly influence the effective mechanical properties andcure shrinkage strains of the composite during cure.

In the present investigation, the curing process of the resin is separated intothree distinct regions for descriptive purposes as shown in Figure 1. In Region I,the resin is in the B-stage condition, fully uncured and assumed to behave as aviscous fluid (negligible stiffness). Region II denotes the curing stage of the resin,where a significant increase in stiffness (chemical hardening) and a reduction inspecific volume (chemical shrinkage) begin to occur. Resin modulus develop-ment is assumed to begin at the point of resin gelation, denoted agm.1’, and isassumed to be complete once the resin is fully cured or diffusion limitation inhib-its further modulus development, at o’~’. The resin chemical shrinkage is as-sumed to behave similarly, with the actual change in specific volume of the resinoccurring until diffusion, ad&dquo;ff (see Figure 1). The bounds during cure for modu-lus development and chemical shrinkage do not, in general, coincide. In fact,they are unique for any given thermosetting system.Region III marks the end of the curing process and no further chemical shrink-

age occurs. In Region III the resin exhibits traditional viscoelastic behavior atelevated temperatures and approaches elastic behavior at lower temperatures.Thermal expansion is the only mechanism contributing to changes in specificvolume in Region III.

This work focuses on the influence the curing process has on the macroscopicinplane stress and deformation development in thick thermosetting composite



629

Figure 1. Resin modulus and chemical shrinkage during cure.



630

laminates. A one-dimensional cure simulation model, based on an incrementaltransient finite difference formulation that accounts for thermal and chemical in-teractions, is developed to predict complex temperature and degree of cure gradi-ents that develop during cure. Material models are proposed to describe the mod-ulus and shrinkage of the resin during cure.Mechanical properties of the fiber phase are assumed independent of cure. A

micromechanics model, employing the resin and fiber constituent behavior, isused to evaluate the instantaneous spatially varying mechanical properties, ther-mal expansion and chemical shrinkage strains within the composite laminate asa function of temperature and degree of cure. Process-induced stress and defor-mation predictions are based on an incremental laminated plate theory model thatincludes temperature gradients, spatially varying cure dependent mechanicalproperties, thermal expansion and chemical shrinkage strains. The incrementallaminated plate theory model and cure simulation analysis are coupled enablingthe prediction of macroscopic inplane process-induced stress and deformation inthick composite laminates during the curing process to be made.The methodology developed for predicting the process-induced stress and de-

formation in thick sections is coded into a FORTRAN computer program. The

program is employed in parametric studies to quantify the influence of processingon residual stress development in thick glass/polyester and graphite/epoxy ther-moset laminates. Model predictions of cure dependent epoxy modulus and curva-ture in unsymmetric graphite/epoxy laminates are correlated with experimentaldata. Processing history (autoclave temperature cycle), laminate thickness andresin cure shrinkage are shown to have a profound influence on the residual stressdevelopment in thick sections. The influence of unsymmetric curing (curing thelaminate from one side) and laminate stacking sequence on process-inducedstress development is also investigated. Results are presented which demonstratethat the traditional assumption of a stress-free laminate temperature is not gener-ally appropriate for predicting residual stresses in thick thermoset compositelaminates.

2. ANALYSIS

2.1 Cure Simulation

The one-dimensional cure simulation analysis presented herein was developedas a simplification of the authors’ full two-dimensional anisotropic cure simula-tion analysis for complex shaped cross-sectional geometries [9,13]. A one-dimensional analysis enables isolation of through-the-thickness processing ef-fects on a fundamental level without the extra computational effort required in atwo-dimensional analysis. The solution strategy utilized in the cure simulationanalysis is consistent with the treatment made by previous investigators [26,27].The principal governing equation utilized in the analysis is the one-dimensionalFourier’s Heat Conduction Equation:



631

The internal heat generation term, 4, accounts for the exothermic chemicalreaction characteristic of thermosetting systems. The parameters kz, e and Cp arethe effective interlaminar (through-the-thickness) thermal conductivity, densityand specific heat of the composite, respectively. These thermal properties are as-sumed constant during the curing process [7]. Total laminate thickness is f. Ab-solute temperature and time are denoted T and t, respectively. The z-direction isnormal to the inplane dimension of a laminate. Thermal properties for glass/polyester and graphite/epoxy used in this study are summarized in Table 1

[26-28] .A generalized temperature boundary condition formulation was employed al-

lowing for either convective, insulated or prescribed temperature boundary con-ditions on the laminate surface:

The temperature and normal derivative of temperature on the laminate surfaceare denoted TS and 8£/8z, respectively. The coefficients keff and hetf representthe effective thermal conductivity and convective heat transfer coefficient on thelaminate surface, respectively. The autoclave temperature cure cycle is incor-porated into the cure simulation through T(t). Prescribed temperature boundaryconditions (k,, = 0 and heff = 1) were employed in all simulations presented inthis work to eliminate the added complexity of interpreting the influence of con-vection on the results [9,13].The internal heat generation term in Equation (1), q, represents the instan-

taneous heat liberated per unit volume of material from the cross-link polymer-ization reaction:

The heat of reaction, Hn is the total heat liberated for complete cure and d a / dtis the instantaneous cure rate. The degree of cure at any time is defined in termsof the instantaneous cure rate through an integral representation:

Table 1. Thermal properties for glass/polyester and graphitelepoxy.



632

The complete description of the cure kinetics for the composite includes thetotal heat of reaction and a description of the rate of reaction as a function of tem-perature and degree of cure. The instantaneous reaction rate is required to com-pute the heat generation [Equation (3)] and degree of cure [Equation (4)] duringthe curing process. Both the total heat of reaction and the reaction rate expressionare typically characterized empirically with isothermal Differential ScanningCalorimetry (DSC) techniques [28]. Reaction rate expressions for glass/poly-ester and graphite/epoxy are different in form due to the inherent differences inthe overall order of the reaction kinetics.The glass/polyester composite contains CYCOM 4102 polyester resin, manu-

factured by the American Cyanamid Corporation, and is reinforced with E-glassfibers (54% by volume). The reaction rate expression for glass/polyester issecond-order overall, (me + ~c = 2), [28] :

The parameter R denotes the universal gas constant. The exponents me and n~, thepre-exponential coefficient, /lc, the activation energy, 0 E~ , and the total heat ofreaction, Hn for glass/polyester are listed in Table 2.The graphite/epoxy composite contains Hercules Corporation’s 3501-6 epoxy

resin and is reinforced with unidirectional AS4 graphite fibers (67% by volume).The reaction rate expression for graphite/epoxy follows a markedly different form[26,27] :

The parameters kl, k2 and k3 are defined by the Arrhenius rate expressions

The pre-exponential coefficients, A1, A2 and A3, the activation energies, AEI,0 E2 and 0 E3, and the total heat of reaction for graphite/epoxy are summarizedin Table 2 [26-28].

Initial conditions for temperature and degree of cure are mathematically repre-sented by



633

Table 2. Cure kinetic parameters forglass/polyester and graphitelepoxy.

T, and a, are the prescribed uniform initial temperature and degree of cure distri-butions within the laminate taken to be the ambient temperature and zero, respec-tively.An incremental transient finite difference technique is employed to solve the

governing equations, boundary and initial conditions that define the cure simula-tion in question. Transient temperature and degree of cure distributions (through-the-thickness) are predicted as a function of the thermal properties, chemical-kinetic parameters, initial and boundary conditions including the autoclavetemperature cure cycle. The laminate is discretized in the thickness dimension (z)with a one-dimensional finite difference nodal grid. The spatial and time deriva-tives in the governing temperature equation [Equation (1)] and the boundary con-dition [Equation (2)] are replaced by their appropriate finite difference approx-imations [9,13]. Transient temperature distributions are solved from the resultingset of finite difference equations using the Alternating Direction Explicit (ADE)Method [29]. Degree of cure distributions are computed from a finite sum ap-proximation of the integral representation given by Equation (4). The solutionstrategy of the cure simulation marches out the autoclave temperature cure cycleuntil its completion. In this manner, the transient temperature and degree of curedistributions through the thickness of the laminate are generated as a function ofthe processing history. Details of the solution strategy are documented elsewhere[9,13].

2.2 Material Models

Two material models are proposed to describe the behavior of the thermosetresin during cure. These models describe the mechanical property changes and



634

volumetric cure (chemical) shrinkage associated with the cross-linked polymer-ization reaction. Property changes and cure shrinkage of the resin during curedirectly influence the homogeneous mechanical properties and process-inducedstrains in the composite. This influence is quantified with a self-consistent fieldmicromechanics model for unidirectional continuous fiber reinforced composites[30,31]. Mechanical property and process-induced strain gradients through thelaminate thickness represent important mechanisms contributing to macroscopicinplane stress and deformation development during cure. The material modelsthat describe the resin behavior during cure, coupling the cure simulation andstress analysis, are discussed below.

2. 2.1 CURE DEPENDENT RESIN MODULUSThe resin modulus model describes the mechanical properties of the resin dur-

ing cure. The resin modulus is strongly cure dependent, influenced by thekinetic-viscoelastic interactions successfully modeled by Dillman and Seferis[32]. While their model was rigorous, independent evaluation of the kinetic andviscoelastic parameters required extensive data reduction procedures. In addi-tion, model predictions outside the temperature range characterized by Dillmanand Seferis were not reliable. Consequently, a more convenient a-mixing rulemodel [33] is proposed here to describe the kinetic-viscoelastic behavior of theresin modulus during cure. The instantaneous isotropic resin modulus, denotedE~, is expressed explicitly in terms of degree of cure:

where

The parameters El- and Em are the assumed fully uncured (Region I) and fullycured (Region III) temperature dependent resin moduli, respectively. The termsCL g.1 and a2bt represent the bounds on degree of cure between which resin mod-ulus is assumed to develop (see Figure 1). The term y is introduced to quantifythe competing mechanisms between stress relaxation and chemical hardening[32]. Increasing y physically corresponds to a more rapid increase in modulus atlower degree of cure before asymptotically approaching the fully cured modulus.Note that this model is extremely sensitive to the kinetics of the resin and, there-fore, the nonisothermal process history [see Equations (5) and (6)]. Results pre-sented in this study assume Em and E1° are constant and y = 0. Unless other-wise noted, it is also assumed that ce-11 I = 0 and ad,ff = 1.The instantaneous resin shear modulus during cure is based on the isotropic

material relation:



635

Poisson’s ratio of the resin, JIm, is assumed constant during cure. As this investi-gation focuses on the influence that the curing process has on stress and deforma-tion development, all mechanical properties of the resin are assumed constantonce cure is complete. Therefore, the analysis predictions for this assumptionmay provide upper bounds on process-induced stresses, since no stress relaxationis considered.The fully uncured and fully cured resin moduli for glass/polyester and graph-

ite/epoxy used in this study are listed in Table 3 [34]. Uncured moduli (E2) arechosen arbitrarily small (negligible stiffness) while the fully cured moduli (E1°)are representative of typical room temperature values.

In Figure 2, the model of Levitsky and Shaffer [21-23] is correlated with thepresent formulation for the chemical hardening of a polyester resin (see Tables 2and 3) during isothermal cure at 100°C. The present model assumes the Poissonratio is constant while the latter model assumes the bulk modulus is constant.Both models predict nearly identical results for modulus development duringcure. Deviation in Poisson ratio is noted at low degree of cure while the modelsconverge rapidly as the resin cures (see Figure 2). The macroscopic compositeproperties, process-induced strains and residual stresses are not significantly in-fluenced by this difference in the Poisson ratio of the resin during cure.

2.2.2 CURE DEPENDENT RESIN CHEMICAL SHRINKAGEA second material model is proposed to describe the volumetric chemical

shrinkage of the resin during cure. Resin shrinkage only occurs during the curingprocess and ceases once cure is complete. Chemical resin shrinkage induces sig-nificant macroscopic strains in the composite, representing an important sourceof internal loading in thick-section laminates in addition to the traditionallyrecognized thermal expansion strains.The volumetric change of a cubic volume element of dimension l, by l2 by 1,

can be expressed in terms of its overall dimensions and the finite dimensionalchanges in three principal directions, All, A/2, A/3, as

An associated change in specific volume, Ov, can be defined in terms of the prin-cipal strain components

Table 3. Resin characteristics during cure.



636

Figure 2. Influence of the constant Poisson ratio assumption on modulus development during cure.

Assuming a uniform strain contraction for all principal strain components, theincremental isotropic shrinkage strain, AE,., of a unit volume element of resinresulting from an incremental specific volume resin shrinkage, Ov&dquo; becomes

The incremental volume resin shrinkage is based on an incremental change indegree of cure, Aa, and the total specific volume shrinkage of the completelycured resin, v.1h, through the following expression:

where

All results presented in this study assume that aot§ = 1. Typical total volumetricresin cure shrinkage values for glass/polyester and graphite/epoxy are listed inTable 3.

2.2.3 CURE DEPENDENT COMPOSITE MECHANICAL PROPERTIESThe effective homogeneous unidirectional mechanical properties of each indi-

vidual lamina within the composite laminate are computed each time increment



637

during the cure simulation. Lamina properties are highly dependent on the fiberand resin constituent properties, and fiber volume fraction. The mechanicalproperties of the resin (except Poisson ratio and thermal expansion which areassumed constant) vary according to the material models presented above. Me-chanical properties of the fiber are assumed constant and independent of cure.During cure, significant composite property gradients develop as a result of thetemperature and degree of cure gradients associated with thick laminate pro-cessing.The self-consistent field micromechanics model [30,31] is used to compute the

instantaneous transversely isotropic mechanical properties and thermal expan-sion coefficients of the lamina (see Appendix). The fiber and resin constituentmechanical properties for glass/polyester and graphite/epoxy are summarized inTable 4 [34]. The 1-direction is coincident with the direction of fiber reinforce-ment. The 2-direction (inplane) and 3-direction (out-of-plane) are both perpen-dicular to the 1-direction. The graphite fiber is assumed transversely isotropicwhile the glass fiber is assumed isotropic. Note the strong resin property de-pendency on degree of cure and thus on the processing history.

Predicted effective transverse and inplane shear moduli of unidirectional glass/polyester are presented in Figure 3 under isothermal cure at 100°C for ceg’.1’values of 0.0 and 0.5 and aili’lf = 1.0. The rapid increase in instantaneous mod-uli is indicative of the rapid curing process and is more pronounced in thea;ef = 0.5 case. Gradients in this type of material response represent a signifi-cant mechanism for inducing stress in thick-section thermosetting laminates dur-ing the curing process.

2.2.4 COMPOSITE CHEMICAL SHRINKAGE STRAINEffective chemical shrinkage strains in the composite are also computed ac-

cording to the micromechanics model and are based on the fiber and resin me-chanical properties, chemical resin shrinkage strain and fiber volume fraction.The inplane principal lamina 1-direction (longitudinal) and 2-direction (trans-verse) chemical shrinkage strain increments, denoted AE 1,1 and Ac- 121, respectively,

Table 4. Fiber and resin constituent mechanical properties.



638

are computed over each time increment in the cure simulation. Specifically, thesestrains are based on the expansional strain relations of the micromechanics modelutilizing the instantaneous mechanical properties of the fiber and resin, the resinchemical shrinkage strain increment, a zero stress-free shrinkage strain in thefiber and the fiber volume fraction (see Appendix).

Volumetric shrinkage of the resin is occurring simultaneously to the effectivemechanical property stiffening effects. Micromechanics predictions of the effec-tive macroscopic composite chemical shrinkage strain account for the combinedinfluence of the volumetric resin shrinkage and instantaneous resin properties.The shrinkage strain in unidirectional glass/polyester under isothermal cure at100°C is illustrated in Figure 4 for a total volumetric resin shrinkage of v5 =6%. While the longitudinal component is small, due to the constraining fiberstiffness, the effective transverse component is signficant, exceeding 1 % . Gradi-ents in these shrinkage strains also represent a significant source for stress devel-opment not recognized in traditional residual stress analyses.

2.2.5 COMPOSITE THERMAL EXPANSION STRAINIncremental thermal expansion strains are also computed over each time incre-

ment during the cure simulation. They are based on the lamina temperature in-crement between two consecutive time steps, AT, and the instantaneous effectivelongitudinal and transverse thermal expansion coefficients, al and az, respec-tively. The incremental longitudinal and transverse strain increments in eachlamina are defined in the usual manner:

2.2.6 TOTAL PROCESS-INDUCED COMPOSITE STRAINThe total stress-free macroscopic process-induced lamina strain increment

(over a single time step) is computed by superimposing the thermal and chemicalstrain contributions. The total process-induced lamina strain increments in thelongitudinal and transverse directions become

Both chemical (shrinkage) and thermal (expansional) strains are inherentlyelongational, consequently no shear strains in the principal material coordinatesystem are required in the analysis. Note, however, that process-induced shearstrains will develop in the global laminate system when other than cross-ply[0/90] laminate constructions are considered.

2.3 Process-Induced Stress ModelingClassical laminated plate theory forms the theoretical basis for the stress analy-

sis presented herein [35,36]. An incremental Hooke’s law formulation is em-



639

Figure 3. Glasslpolyester moduli during Isothermal cure at 100°C.

--..- .........- B.&dquo;&dquo;..’I8~’’’’’&dquo;’’’1

Figure 4. Glasslpolyester chemical shrinkage strain during isothermal cure at 1 aaoe.



640

ployed which utilizes the instantaneous effective composite properties and incre-mental process-induced lamina strain increments as input. Various incrementallaminated plate theory solution strategies and convergence criteria are discussedelsewhere [37]. The theory and methodology employed in this study are dis-cussed below.

Prior to the stress and deformation calculations for a single time step incre-ment, the cure simulation is performed to yield the temperature and degree ofcure distributions within the laminate. From these distributions, the instan-taneous effective mechanical properties and incremental process-induced strainsin the laminate are computed based on the material models discussed above.Stress and strain increments are then computed from the plate theory analysisover each time step in the cure simulation. This procedure is repeated for eachtime step increment of the cure simulation. By maintaining a cumulative sum ofthe stress and strain increments, a complete description of their transient devel-opment during the curing process and subsequent cool-down to uniform ambienttemperature is obtained.The laminate is discretized with nodal points in the z-direction, consistent with

the finite difference grid employed in the cure simulation analysis. Mesh spacingis established such that a ply is defined between each pair of nodes in the finitedifference grid, uniformly spaced at distance Az. The temperature and degree ofcure of a ply is taken as the arithmetic average of the cure simulation predictedvalues at the top and bottom nodes of that ply.The theoretical basis for the analysis employed in this study is laminated plate

theory [35,36]. This analysis addresses macroscopic inplane stress developmentinduced by processing gradients in the out-of-plane direction. The analysis is in-cremental in time and is performed at discrete time steps during the cure history.Transient processing stresses and strains are computed as the cumulative total ofall previous increments to a given point in the cure cycle history. The analysis isgeneral in nature and therefore accommodates laminates of arbitrary stacking se-quence.The effective laminate inplane force and moment resultants are first calculated

for the time step increment. The incremental inplane force resultants, ANT, AN§and 0 N y , and incremental moment resultants, 0 Mx , 0 My and AM$, aregiven by:



641

The total number of plies in the laminate is N. The matrix, Q ;, is the trans-formed plane stress stiffness matrix of the kth ply defined in the global (x, y) lami-nate coordinate system in the traditional manner [35,36]. The stress-free incre-mental ply strains, AE 1, ~Ey and dE;y, defined in the global laminate coordinatesystem, are then computed based on the incremental stress-free process-inducedply strains in the principal lamina coordinate system [Equation (16)] through asecond-order tensoral strain transformation.The instantaneous effective laminate compliance coefficients are computed in

the traditional manner [35,36], with spatial variation of instantaneous mechanicalply properties taken into account. The effective incremental laminate inplanestrains, DEX, DEY and DEX,,, and curvatures, 0 xx, dxy and A xxy are then evalu-ated as the product of the laminate compliance matrix and the process-inducedeffective laminate inplane force and moment resultants. Once the incremental in-plane strains and curvatures are determined, the laminate ply strain increments,A~, DEY and dExy are computed through the classic strain-displacement relations[35,36].The incremental ply stresses are based on the difference between the laminate

ply strain increments and the stress-free process-induced ply strain incrementsthrough the following expression:

Transient process-induced stress and strain distributions through the laminatethickness are obtained from the cumulative sum of the increments computed ateach time step during the cure simulation. The governing equations for the curesimulation, material models and stress analysis were coded into a FORTRANcomputer program. A flow diagram summarizing the key steps in the solutionstrategy is presented in Figure 5. As an example of the computational efficiencyof the computer simulation, a 2.54 cm thick laminate cured for six hours, con-sisting of 30 nodes and 10,000 time steps, takes approximately two hours of CPUtime. Parametric studies were performed to gain fundamental insight into

process-induced stress development in thick-section thermoset laminates. Resultsare presented and discussed in the next section.

3. RESULTS AND DISCUSSION

In this section input for the process-induced stress model is first summarized.Predictions of the effective composite inplane modulus and process-inducedstrain development, associated with the cure reaction of the resin, are presented.Several aspects of the analysis are then validated by comparison with experimen-tal results of modulus and laminate curvature found in previously published liter-ature. Parametric studies are presented that focus on the effects of the autoclave



642

Figure 5. Process-Induced stress modeling flow diagram.



643

temperature cure cycle, laminate thickness, stacking sequence, and resin cureshrinkage on process-induced stress and deformation development.

3.1 Input Parameter Summary

Required model input includes thermal properties, kinetic parameters and me-chanical properties. This input for glass/polyester and graphite/epoxy was sum-marized in Tables 1 through 4 in the previous section. Typical autoclave tempera-ture cure cycles for glass/polyester and graphite/epoxy employed in the many ofthe parametric studies presented in this work are illustrated in Figure 6. Addi-tional input (specified where appropriate) includes total laminate thickness,stacking sequence and volumetric resin shrinkage.As with all numerical solution techniques, the potential for computational

round-off error exists. Convergence studies were conducted to evaluate the sen-sitivity of the analysis to nodal spacing and time step size increment. A nodalspacing of 2.30 x 10-3 m and a time step of 1.0 second were found to yield con-verged solutions for laminates up to 7.62 cm thick and were used to generate allresults presented in this investigation.

3.2 Comparison with Experimental Results

Hahn and Kim subjected several graphite/epoxy (T300/3501-6) unidirectionaland [0/90] unsymmetric laminates to interrupted cure cycles to monitor the devel-opment of transverse modulus (in the unidirectional laminates) and curvature in-duced by cure shrinkage and thermal strains (in the unsymmetric laminates) [38].Laminates were approximately 1 mm in thickness and the cure cycles followedthe manufacturer’s recommended cure cycle (see Figure 6). At each of a series ofpoints during cure, a unidirectional and a cross-ply laminate were cooled to roomtemperature at 3°C/min. The unidirectional laminates were tensile tested to de-termine the transverse modulus, and the deflections of the unsymmetric laminateswere measured to determine process-induced curvatures. Figure 7 shows ex-cellent agreement between the measured and predicted transverse modulus devel-opment based on the material model presented herein. Note that the horizontalaxis refers to time of cure stop. This indicates the time at which the cure cycle wasinterrupted and cooling to room temperature initiated, and does not indicate thetime of the full cycle to which the laminate was subjected as not all laminateswere cooled from the same temperature.

Figure 8 shows the development of nondimensionalized curvature (with respectto laminate thickness) as a function of the time of cure stop for total epoxy resinvolumetric shrinkages of 1.0 and 1.5 %. Again we see good agreement betweenmeasured and predicted values. Significant over-prediction of the curvature earlyin the cycle is thought to arise from the fact that the highly viscoelastic behaviorof the resin, which has not yet reached gelation, is not incorporated in the presentmodel. Also, the fracture surfaces of the tensile tested specimens in the region ofthe curvature over-prediction exhibited little or no interfacial bonding betweenthe fibers and the resin, leading to a relatively unconstrained resin and reducedstress development. By defining the gel point as the point at which the viscosity



Figure 6. Typical autoclave cure cycles for glasslpolyester and graphitelepoxy.

Figure 7. Experimental vs. predicted composite transverse modulus development in a uni-directional graphitelepoxy laminate.

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Figure 8. Experimental vs. predicted curvature during cure of an unsymmetric (gO/D)graphitelepoxy laminate.

of the resin reaches 100 Pa-sec, Hahn and Kim note that the curvature and thetransverse modulus both increase sharply immediately after the gel point, possi-bly indicating that the development of residual stress can be significantly affectedby the mechanical properties of the resin before full cure has been reached. Goodagreement with the experimental results provides confidence in the accuracy ofthe analysis for predicting stress development during the curing process.

3.3 Thickness EfFects

To gain an appreciation for the mechanisms introduced in this investigation,complex temperature and degree of cure distributions are illustrated in unidirec-tional glass/polyester laminates of varying thickness exposed to the same curecycle. The glass/polyester cure cycle, indicated in Figure 6, yields a maximumtemperature exotherm in a 2.54 cm thick laminate at 164 minutes. Figure 9 illus-trates the influence of laminate thickness on temperature distributions at 164minutes into this cure cycle. Temperature exotherm increases with laminatethickness below 2.54 cm while the 5.08 cm laminate is still heating. Correspond-ing degree of cure distributions are illustrated in Figure 10. Degree of cure gradi-ents are most severe at this point in the 2.54 cm laminate. The interior of the 2.54cm laminate is essentially cured. In contrast, the 5.08 cm laminate is essentiallyuncured at the interior. At a later point in the cure cycle, the thicker laminate willexotherm and its distributions will reverse shape similar to the thinner laminates,ultimately developing more severe gradients. These results demonstrate the com-plex temperature and degree of cure gradients which develop during the curing



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Figure 9. Temperature distributions in glass/polyeshr laminates at 164 minutes.

Figure 10. Degree of cure distributions in glasslpolyester laminates at 164 minutes.



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process of thick thermoset laminates. These gradients in turn have a profound in-fluence on the evolution process-induced stress and deformation.

Residual process-induced transverse stress distributions in 1.38, 1.85 and 2.54cm thick unidirectional glass/polyester laminates were predicted for the cure cy-cle shown in Figure 6 with vl = 6%. The results are presented in Figure 11.These significant stress distributions remain after complete cure and uniform am-bient temperature conditions within the laminate exist. The parabolic stress pro-files are self-equilibrating tensile stresses at exterior and compressive stresses atthe interior regions. This profile is unique to laminates that cure predominantlyfrom the inside to the outside surface of the part. Interior regions curing first, dueto the exotherm, are less constrained by the uncured exterior region duringshrinkage. As the exterior cures and contracts, it is constrained by the cured in-terior region of appreciable stiffness and the process-induced residual stressesshown in Figure 11 develop. Stress gradients are accentuated with increasing lam-inate thickness. For comparison purposes, it is pointed out that stress analysesbased upon the traditional sources of residual stress associated with laminate

stacking sequence predict that unidirectional laminates are stress free. This isclearly not the case with thick laminates.

Residual process-induced transverse stress distributions in unidirectional

glass/polyester laminates 2.54, 5.08 and 7.62 cm thick, utilizing the same curecycle and resin shrinkage, are illustrated in Figure 12. A dramatic parabolic stressreversal of the distributions in the 5.08 and 7.62 cm laminates are predicted wherethe interior regions are in tension and the exterior regions are in compression.This reversal results from the change to an outside to inside cure history in the

Figure 11. Residual process-induced inplane transverse stress distributions (f s 2.54 cm).



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Figure 12. Residual process-induced inplane transverse stress distributions (f a 2.54 cm).

thicker laminates and clearly demonstrates the significant influence the laminatethickness has on the evolution of process-induced stress. The magnitude of peakstresses and gradients which develop are significant enough to initiate transversecracks [3-6].

3.4 Influence of Resin Modulus DevelopmentThe development of process-induced stress in thick thermoset laminate is

strongly influenced by both the resin modulus and chemical shrinkage develop-ment. The interactions between these two phenomena will depend on the particu-lar thermosetting system and will have a profound influence on process-inducedstress distributions. In the following example, the influence of a~ on process-induced residual stress distributions is presented for the 2.54 cm (1.0 inch) lami-nate presented above. In Figure 13 residual process-induced inplane transversestress distributions are presented for a~ef values of 0.0, 0.4, and 0.6. Here themore rapidly increasing modulus development for am’ = 0.4 and 0.6 (seeFigure 3) have resulted in more severe residual stress distributions. This exampledemonstrates that the interactions between chemical shrinkage and resin modulusdevelopment during cure can significantly influence process-induced stress dis-tributions in thick thermosetting laminates.

3.5 Autoclave Temperature Cure Cycle Effects

The influence of the autoclave temperature cure cycle history on stress develop-ment is investigated by considering the curing of a unidirectional 2.54 cm



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glass/polyester laminate exposed to the temperature histories illustrated in Figure14 and a resin shrinkage of v5 = 6%. Resulting residual process-induced trans-verse stress distributions for temperature ramps of 0.25, 0.5 and 1.0 (°C/minute)are presented in Figure 15. The stress distributions developing under the tempera-ture ramps of 0.25, 0.5 and 1.0 (°C/minute) induce compressive stresses at the in-terior, while the more rapid curing process results in the development of tensileinterior stresses. These results demonstrate the importance of the processinghistory on residual stress development in thick laminates. The slower thermalramps create an inside to outside cure history and consequently develop internalcompressive stresses. As the ramp is increased the cure history changes to an out-side to inside cure, resulting in a reverse in the parabolic stress distribution.

3.6 Cure Shrinkage Effects

The influence of resin shrinkage on residual stress development was also inves-tigated. Total volumetric resin shrinkages, v5, of 0, 1, 3 and 6 % are assumed inthe simulations of a 2.54 cm unidirectional glass/polyester laminate cured ac-cording to the cure cycle in Figure 6. Residual transverse stress distributions areplotted against dimensionless thickness in Figure 16.The plots indicate that the magnitude of the resin shrinkage significantly in-

fluences the parabolic stress distribution. The profile for v5 = 0 % isolates the

Figure 13. Influence of « 9e°~ on residual process-induced inplane transverse stress distri-butions.



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Figure 14. Glasslpolyester autoclave temperature cure cycle ramps.

Figure 15. Glasslpolyester autoclave temperature cure cycle ramp effects on transversestress distributions (f = 2.54 cm).



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- - - - - - - -- - - -

,- - -,

Figure 16. Resin shrinkage effects on residual inplane transverse stress distributions in uni-directional glasslpolyester laminates (f = 2.54 cm).

effect of nonuniform curing without resin shrinkage. It is interesting to note thatfor v5 = 1 % the laminate is essentially stress free. In this case the processingstrains due to chemical shrinkage are effectively equal to magnitude but oppositein sign to the thermal expansion processing strains.

3.7 Unsymmetric Curing Effects

All simulations presented to this point exhibited symmetric curing about themidplane of the laminate. The processing histories imposed on the laminates in-duced either inside-to-outside or outside-to-inside curing patterns. Consequently,all residual stress profiles were shown to be symmetric about the laminate mid-plane. When the thermal history is unsymmetric, the resulting stress profiles willalso be unsymmetric. The following example demonstrates the influence of un-symmetric curing on residual stress development.

Unidirectional glass/polyester laminates of 1.38, 2.54 and 3.39 cm thicknesswere subjected to unsymmetric boundary conditions. The autoclave temperaturecure cycle illustrated in Figure 6 was prescribed on the bottom surface of the lam-inate. An insulated boundary condition was imposed on the top surface. A volu-metric resin shrinkage of v5 = 6% was assumed. In these simulations, the curesweeps from the bottom to the top surfaces. The resulting residual transverse



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stress distributions are illustrated in Figure 17. Similar gradients are predicted forlongitudinal stress distributions. Note that the stress profiles are self-equilibrat-ing. It is interesting to contrast the magnitude of the peak transverse stress in the2.54 cm laminate presented here to that which developed in the symmetricallycured laminate presented in Figure 11. The symmetric curing process resulted intransverse stresses ranging between - 8.5 and 12.0 MPa while the unsymmetriccuring process reduced stresses substantially ( -2.0 and 2.0 MPa). This result isnot surprising since the unsymmetric cure induces warpage in the laminate, re-ducing the magnitude of the inplane residual stress.

3.8 Stacking Sequence Effects

The parametric studies presented to this point have focused on the unidirec-tional laminate configuration. This enabled the mechanisms of spatial solidifica-tion to be isolated, without the influence of the traditionally recognized stackingsequence effects. In the following examples, the influence of processing on stressdevelopment in [0/90] glass/polyester cross-ply laminates is quantified.

In the first example, a 2.54 cm laminate of [0/90] symmetric construction with8 repeating units, ([0/90]..,), was cured symmetrically according to the glass/polyester cure cycle with v5 = 6.0% (see Figure 6). The resulting transversestress distributions are shown in Figure 18. Such stress profiles are similar towhat would be predicted by traditional laminated plate theory analysis that

Figure 17. Influence of unsymmetric curing on transverse residual stress profiles in uni-directional glass/polyester laminates.



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’&dquo;_&dquo;&dquo;~._&dquo;V- _,I_VV B.YI. -,

Figure 18. Transverse residual stresses in a [0/90Jas laminate.

assumes a stress-free temperature. In this simulation, spatial solidification is

present but negligible and one concludes that stacking sequence effects dominate.In thicker laminates, however, spatial solidification cannot be neglected. Con-

sider the resulting stress profiles in symmetric [0/90]ns laminates with n = 8, 16and 24 (corresponding to total laminate thicknesses of 2.54, 5.08 and 7.62 cm, re-spectively) shown in Figure 19. The profiles represent the envelope of peak nor-mal stresses in the 0° or 90° plies through the laminate thickness. The stressesare discontinuous across ply interfaces and are equal and opposite in magnitude.Processing is shown to substantially increase the stress gradient with increasinglaminate thickness. In contrast to the 2.54 cm laminate, the process-inducedstress in the thicker laminates exhibit significant gradients through the thicknessthat are also much greater in magnitude in comparison with the traditional stack-ing sequence stress. This result further illustrates the significant influence theprocessing history has on stress development in thick-section thermosets, evenfor laminates of arbitrary stacking sequence.

3.9 Stress-Free TemperatureTraditional residual stress predictions are based on a uniform stress-free tem-

perature, commonly assumed to be the cure temperature. The uniform tempera-ture drop to the operating temperature defines the temperature difference used tocompute residual stresses. In the following example, the influence of resin

shrinkage on the assumed stress-free temperature of a [0/90]~ glass/polyesterlaminate is quantified.



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The laminate thickness is chosen sufficiently thin to minimize temperature anddegree of cure gradients during processing. In this case, stacking sequence is thedominant source of residual stress. The autoclave cure cycle is defined in Figure6. Ply stresses are predicted for volumetric resin shrinkages between 0 and 6 % .An equivalent stress-free temperature is defined from our predictions for com-parison with the traditional methodology based on the assumption that AT =106°C (the difference between ambient and the cure temperature). A dimen-sionless stress-free temperature difference is defined as the ratio of the predictedto the assumed temperature change. In Figure 20, dimensionless stress-free tem-perature is plotted against the volumetric resin shrinkage. Only for a volumetricshrinkage of 1 % is the traditional assumption valid (~ T = -106 °C) . The resultsdemonstrate the significant influence that resin shrinkage can have on theassumed stress-free temperature, and thus the magnitude of the resulting residualstress distributions.Another point worth noting is that in our simulation, the evolution of stress

throughout the curing process is predicted and therefore no stress-free tempera-ture assumption is required. Furthermore, the severe gradients in temperatureand degree of cure that were shown to develop in thick laminates suggest that thetraditional approach is not appropriate. This is apparent from the fact that signifi-cant residual stresses were shown to develop in thick unidirectional laminates.Recall that the traditional analysis predicts that unidirectional laminates are stressfree. Therefore, the traditional residual stress analyses are not appropriate forthick-section laminates in general. The appropriateness of the assumption de-

Figure 19. Normal residual stress envelopes in [O/901ns laminates.



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Figure 20. Influence of resin shrinkage on dimensionless stress-free temperature.

pends on many factors such as laminate thickness, resin shrinkage and the pro-cessing history.

4. CONCLUSIONS

A fundamental study of process-induced stress and deformation in thick-section thermosetting composite laminates was presented. A methodology wasproposed for predicting residual stress development during the curing processthat does not require a stress-free temperature to be assumed. A one-dimensionalcure simulation analysis was coupled to an incremental laminated plate theorymodel to study the relationships between complex gradients in temperature anddegree of cure, and process-induced residual stress and deformation.

Material models were proposed to describe the mechanical properties, thermaland chemical strains of the thermoset resin during cure. The material modelswere incorporated into a micromechanics model to predict the effective mechani-cal properties and process-induced strains of the composite during cure. Thermalexpansion and cure shrinkage contribute to changes in material specific volumeand were shown to represent important sources of internal loading included in theanalysis. The point during cure when resin modulus begins to develop gel )was shown to have a significant influence on the magnitude of process-inducedresidual stress that can develop in thick thermoset laminates. Specifically, how-ever, it is the interactions (overlap) between resin modulus (a;ef ::5 a ::5 a%W)and chemical shrinkage (0 <_ a <_ ad;ff) during cure that will influence pro-



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cess-induced residual stress development. Although this was not explored in thiswork, the topic offers an important future research opportunity.Temperature and degree of cure gradients that develop during the curing pro-

cess induce mechanical property and process-induced strain gradients in thecomposite laminate. These gradients represent important mechanisms that con-tribute to stress and deformation development not previously considered in tradi-tional residual stress analyses of laminated composites. Model predictions ofcure dependent epoxy modulus and curvature in unsymmetric graphite/epoxylaminates were found to be in good agreement with experimental data presentedin previously published literature.The effects of processing history (autoclave temperature cure cycle), laminate

thickness, resin cure shrinkage and laminate stacking sequence on the evolutionof process-induced stress and deformation in thick-section glass/polyester andgraphite/epoxy laminates during cure were quantified. Processing of unidirec-tional laminates was investigated to provide fundamental insight into the impor-tant mechanisms governing residual stress development during cure. While tradi-tional sources of residual stress associated with laminate stacking sequencepredict that unidirectional laminates are stress free, our results clearly indicatethe potential for significant process-induced stress development in thick thermo-set laminates.

Residual stresses were shown, in general, to increase with increasing laminatethickness, autoclave temperature ramp and resin shrinkage. Unsymmetric curingwas shown to significantly reduce residual stress in comparison with a symmetriccuring process. Spatial solidification and stacking sequence effects were found tobe equally important in thick-section cross-ply laminates. Furthermore, theresults indicated that the assumption of a stress-free temperature for residualstress predictions is not appropriate for thick sections in general.The present study has demonstrated fundamental mechanisms that contribute

to residual stress development that have not been previously considered in tradi-tional residual stress analyses of laminated composites. The significant magni-tude of process-induced residual stress identified in this work substantiates theprevalent concerns associated with processing difficulties of thick thermosettinglaminates, such as delaminations and matrix cracking. The results clearly in-dicate that the mechanics and performance of thick-section thermoset laminatesare strongly dependent on processing history.

ACKNOWLEDGEMENTS

This work was funded by the Army Research Office University Research Initia-tive Program (Dr. Andrew Crowson, Director of the Material Science Division).The authors are grateful for their financial support. We are also grateful for thesupport of the Ballistic Research Laboratory (Dr. Bruce Burns, Chief of the Me-chanics and Structures Branch, Interior Ballistic Division).

APPENDIX A: CONTINUOUS FIBER MICROMECHANICS MODEL

The self-consistent field model approach for unidirectional continuous fiber re-inforced composites [30,31] offers a more realistic approximation of the effective



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lamina properties than mechanics of materials models based on the uniformstress or strain assumption. The model also offers mathematically tractableclosed-form expressions for the effective lamina mechanical properties and ex-pansion strains. Theoretical background of the model is presented in greaterdetail elsewhere [30,31].

In the equations presented below, the subscripts 1, 2, and 3 refer to the direc-tions in the principal coordinate system of the lamina. Subscripts m and f corre-spond to the matrix (resin) and fiber properties, respectively. Fiber volume frac-tion of the lamina is denoted as vf and k is the isotropic plane strain bulk modulusdefined by

A.I Engineering Constants

The following equations define the transversely isotropic engineering constantsof the lamina. The longitudinal Young’s modulus:

The major Poisson’s ratio:

The inplane shear modulus:

The transverse shear modulus:

The transverse Young’s modulus:



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where k, is the effective plane strain bulk modulus of the composite given by

The transverse Poisson’s ratio:

A.2 Expansional Strains

Expansional strain expressions are used to evaluate the inplane chemicalshrinkage strains and the effective coefficients of thermal expansion of the lamina(«1 = E, and a2 = E2). The following equations define the transversely isotropicexpansional strains of the lamina. The longitudinal direction expansional strain:

The transverse direction expansional strain:

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process-induced stress and deformation in thick-section thermoset composite laminates

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